June 2025 Paper 3 Q3
3. A manufacturer makes components for the car industry.
On average 1.5% of the components made are defective.
Once every six months, the manufacturer tests whether the proportion of defective components has changed using hypotheses \(\mathrm{H}_0 : p = 0.015\) and \(\mathrm{H}_1 : p \neq 0.015\)
A random sample of 260 components is taken and 8 are found to be defective.
| Scheme | Marks | AO |
|---|---|---|
| [Let \(D\) = the number of defective components] \(D \sim \mathrm{B}(36, 0.015)\) | M1 | 3.3 |
| (i) \([\mathrm{P}(D = 2) =]\ 0.084792\ldots\) awrt 0.0848 | A1 | 1.1b |
| (ii) \([\mathrm{P}(D > 3) = 1 - \mathrm{P}(D \leqslant 3) = 1 - 0.9979\ldots =]\ 0.002032\ldots\) awrt 0.00203 | A1 | 1.1b |
| (3) |
Notes
M1: for selecting \(\mathrm{B}(36, 0.015)\) sight of or use of.
Must be \(\mathrm{B}(\ldots)\) [Implied by sight of \(\mathrm{Po}(0.54)\)]
May be implied by sight of: 0.0848 or awrt 0.085 or 0.00203 or 0.998 or 0.002
(i) A1: for awrt 0.0848
(ii) A1: for awrt 0.00203 [Allow \(2.03 \times 10^{-3}\)]
Use of Poisson \(\mathrm{Po}(0.54)\) (i) 0.08496 scores A0 (ii) 0.002309… awrt 0.00231 scores A1
Notation: We do not accept “calculator speak” e.g. \(N = 36\), \(p = 0.015\) instead of \(\mathrm{B}(36, 0.015)\)
| Scheme | Marks | AO |
|---|---|---|
| [Let \(X\) = the number of defective components] \(X \sim \mathrm{B}(260, 0.015)\) | M1 | 3.3 |
| \([\mathrm{P}(X \geqslant 8) =]\ 0.0441(1\ldots)\) | A1 | 1.1b |
| [\(0.04 > 0.025\) so not significant] Insufficient evidence that proportion/probability of defective components has changed (o.e.) | A1 | 2.2b |
| (3) | ||
| (6 marks) |
Notes
Notation: We do not accept “calculator speak” e.g. \(N = 36\), \(p = 0.015\) instead of \(\mathrm{B}(36, 0.015)\)
M1: for sight of \(\mathrm{B}(260, 0.015)\)
may be implied by probability of awrt 0.0441 or awrt 0.0264 or awrt 0.956
or correct CR \(X = 0\) [ or \(\leqslant 0\)] and \(X \geqslant 9\) or \(\mathrm{P}(X \geqslant 8) = 0.04\) or better
A1: for probability of awrt 0.0441 (selected if more probabilities seen)
Condone \(\mathrm{P}(X = 8) = 0.0441\ldots\)
If \(\mathrm{P}(X \geqslant 8)\) is explicitly seen allow 0.04 or better
or for the correct CR \(X = 0\) [ or \(\leqslant 0\)] and \(X \geqslant 9\). Allow acceptance region [1, 8] (o.e.)
A1: dep on M1A1 for a correct conclusion in context.
Allow e.g:
proportion/probability of defective components has not changed
proportion/probability of defective components is still 0.015
Allow “defectives” instead of “components”
If a comparison is seen (other than \(0.956 < 0.975\)) it must be with 0.025
Incorrect comparisons score A0 (condone missing comparison)
Do not accept contradictory statements e.g. “significant so insufficient evidence of change in proportion” would score A0
“Insufficient evidence that \(p \neq 0.015\)” is A0, we need the words.
SC: Poisson Approximation (If you see \(\mathrm{Po}(3.9)\) or probability 0.0454…)
B2 and mark as M0A1A1
B1: for probability of awrt 0.0454 or 0.04… if \(\mathrm{P}(X \geqslant 8)\) seen or CR as above
B1: for correct conclusion in context (same rules as above)